This document discusses mathematical modeling of mechanical systems involving translational and rotational motion. It explains how to form differential equations of motion using Newton's laws and analogies to electrical systems. Models with multiple degrees of freedom are addressed by considering the independent motion of individual points/components and summing the relevant forces for each. Examples of 2 and 3 degree of freedom systems are presented for both translation and rotation.

1. ME 176
Control Systems Engineering
Department of
Mechanical Engineering
Mathematical Modeling

2. Mathematical Modeling: Mechanical Systems
Translational Motion
- Analogous to electrical variables
and components.
- Use of Newton's Law to form
differential equation of motion.
Assumptions:
- A positive direction of motion.
- Sum forces equal to Zero.
- Initial conditions of Zero.
Department of
Mechanical Engineering
Components:
- fv is coefficient of viscous friction .
- M is mass .
- K is spring constant.
- f(t) is force.
- x(t) is displacement.

3. Mathematical Modeling: Mechanical Systems
Translational Motion
Department of
Mechanical Engineering
for Analogy for Solving

4. Mathematical Modeling: Mechanical Systems
Translational Motion
Case 1 :
Force =>Voltage Velocityt =>Current Displacementt =>Charge
Spring => Capacitor Mass => Inductor Damper => Resistor
Summing forces in terms of velocity => Mesh equation
Department of
Mechanical Engineering

5. Mathematical Modeling: Mechanical Systems
Translational Motion
Case 2 :
Force =>Voltage Velocityt =>Current Displacementt =>Charge
Spring => Inductor Mass => Capacitor Damper => Resistor
Summing forces in terms of velocity => Nodal equation
Department of
Mechanical Engineering

6. Mathematical Modeling: Mechanical Systems
Translational Motion
Department of
Mechanical Engineering

7. Mathematical Modeling: Mechanical Systems
Translational Motion
Department of
Mechanical Engineering
Mass
Damper
Spring

8. Mathematical Modeling: Mechanical Systems
Translational Motion
1. Number of equations of motion to describe system is equal to number of
linearly independent motions.
2. Linearly independent motion is a point in a system that can move even if
all other points are still.
3. Linearly independent motions is also called "degrees of freedom."
Solving Equations with High Degrees of Freedom ( > 1):
- Draw free body diagram for each point representing a linearly
independent motion.
- For each analysis, assume other points are still.
- For each analysis, consider only forces related to motion of point.
- For each analysis, use Newton's law and sum all forces to zero.
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Mechanical Engineering

9. Mathematical Modeling: Mechanical Systems
Translational Motion : 2 Degrees of Freedom
Department of
Mechanical Engineering

10. Mathematical Modeling: Mechanical Systems
Translational Motion : 2 Degrees of Freedom
Department of
Mechanical Engineering

11. Mathematical Modeling: Mechanical Systems
Translational Motion : 2 Degrees of Freedom
Department of
Mechanical Engineering

12. Mathematical Modeling: Mechanical Systems
Translational Motion :
3 Degrees of Freedom
Department of
Mechanical Engineering

13. Mathematical Modeling: Mechanical Systems
Translational Motion :
3 Degrees of Freedom
Department of
Mechanical Engineering

14. Mathematical Modeling: Mechanical Systems
Translational Motion :
3 Degrees of Freedom
Department of
Mechanical Engineering

15. Mathematical Modeling: Mechanical Systems
Translational Motion :
3 Degrees of Freedom
Department of
Mechanical Engineering

16. Mathematical Modeling: Mechanical Systems
Rotational Motion
- Same as rotational just that
torque, replaces force; and
displacement is measured
angular.
- Analogous to electrical variables
and components.
- Use of Newton's Law to form
differential equation of motion.
Assumptions:
- A positive direction of motion.
- Sum forces equal to Zero.
- Initial conditions of Zero.
Department of
Mechanical Engineering
Components:
- D is coefficient of viscous
friction .
- J is moment of inertia .
- K is spring constant.
- T is Torque.
- is angular displacement.

17. Mathematical Modeling: Mechanical Systems
Rotational Motion
Department of
Mechanical Engineering
for Analogy for Solving

18. Mathematical Modeling: Mechanical Systems
Rotational Motion
Case 1 :
Torque =>Voltage Velocitya =>Current Displacementa =>Charge
Spring => Capacitor Inertia => Inductor Damper => Resistor
Summing forces in terms of velocity => Mesh equation
Department of
Mechanical Engineering

19. Mathematical Modeling: Mechanical Systems
Rotational Motion
Case 2 :
Torque =>Voltage Velocitya =>Current Displacementa =>Charge
Spring => Inductor Mass => Capacitor Damper => Resistor
Summing forces in terms of velocity => Nodal equation
Department of
Mechanical Engineering

20. Mathematical Modeling: Mechanical Systems
Rotational Motion : 2 Degrees of Freedom
Department of
Mechanical Engineering

21. Mathematical Modeling: Mechanical Systems
Rotational Motion : 2 Degrees of Freedom
Department of
Mechanical Engineering

22. Mathematical Modeling: Mechanical Systems
Rotational Motion : 2 Degrees of Freedom
Department of
Mechanical Engineering

23. Mathematical Modeling: Mechanical Systems
Rotational Motion : 3 Degrees of Freedom
Department of
Mechanical Engineering

24. Mathematical Modeling: Mechanical Systems
Rotational Motion :
3 Degrees of Freedom
Department of
Mechanical Engineering

25. Mathematical Modeling: Mechanical Systems
Rotational Motion :
3 Degrees of Freedom
Department of
Mechanical Engineering

26. Mathematical Modeling: Mechanical Systems
Rotational Motion :
3 Degrees of Freedom
Department of
Mechanical Engineering